A Modi…ed Mann Algorithm For Solving Convex Minimization And Fixed Point Problems With Composed Nonlinear Operators

A Modi…ed Mann Algorithm For Solving Convex Minimization And Fixed Point Problems With Composed Nonlinear Operators

Thierno Mohamadane Mansour Sow

, Ngalla Djitte

, Yahya Baba El Yekheir

Received 12 November 2019

Abstract

The main aim of this paper is to introduce and study an iterative algorithm, which is based on the Krasnoselskii-Mann iterative method and the gradient-projection algorithm for solving a constrained convex minimization problem and …xed point problem with quasi-nonexpansive and …rmly nonexpansive mappings in a real Hilbert space. Finally, we apply our main result for …nding a common solution of convex minimization problem, …xed point problem and equilibrium problem. Essentially, a new approach for solving some nonlinear problems is provided.

1 Introduction

LetH be a real Hilbert space with inner producth; iand induced normk:k: LetK be a nonempty, closed and convex subset ofH: Consider the following constrained convex minimization problem:

miny2Kg(y); (1)

where g : K ! R is a convex function. Assume that (1) is consistent (i.e., it has a solution) and we use to denote its solution set. It is well known that the gradient-projection algorithm (GPA, for short) is usually applied to solve the minimization problem (1). This algorithm generates a sequencefx_ng through the recursion:

x_n+1=P_K(x_{n n}rg(x_n)); n 0; (2)

where the initial guessx02Kis chosen arbitrarily andf ⁿgis a sequence of stepsizes which may be chosen in di¤erent ways. GPA (2) has well been studied in the case of constant stepsizes n = for alln (see the books [27, 28]). A fundamental convergence result for GPA (2) is the following one which can be found in literature (cf. [ [28], Theorem 6.1] with constant stepsize).

Theorem 1 ([28]) Let fxng be the sequence generated by GPA(2). Assume (i) g is continuously di¤ erentiable and its gradient is Lipschitz continuous:

krg(x) rg(y)k Lkx yk; 8x; y2Rⁿ; where L 0is a constant;

(ii) the setK0:=fx2K:g(x) g(x0)g is bounded;

(iii) the sequencef ng satis…es the condition:

0<lim inf

n!1 n lim sup

n!1 n < 2 L:

Mathematics Sub ject Classi…cations: 47H05, 47J25.

yAmadou Mahtar Mbow University, Dakar Senegal

zDepartment of Mathematics, Gaston Berger University, Saint Louis Senegal

xDepartment of Mathematics, Gaston Berger University, Saint Louis Senegal

462

Then, the sequence fx_ng generated by the gradient-projection algorithm (2) converges weakly to a solution of (1).

However, Xu [33] constructed a counterexample which shows that algorithm (2) does not converge in norm in an in…nite-dimensional space, and also presented two modi…cations of gradient-projection algorithms which are shown to have strong convergence.

In 2012, H. Iiduka [19] introduced the following algorithm for solving problem

Algorithm 1 Step 0. Choosex₀2H arbitrarily, set ₀ (0;1); ₀ (0;1)andd₀= rg(x₀)arbitrarily and letn:= 0:

Step 1. Given. xn 2H anddn2H, choose n (0;1); n (0;1)and compute xn+12K as yn=T(xn+ ndn);

x_n+1= _nx₀+ (1 _n)y_n:

Step 2. Choose _n+12(0;1]and compute the directiond_n+1 2H, by dn+1= rg(xn) + _n+1dn: Update n:=n+ 1and go to Step 1.

Under suitable conditions, he proved thatfxngⁿ2Nin Algorithm1weakly converges to a unique solution to Problem (1). Recently, studies on solutions of the minization problem (1) were extensively carried out in Hilbert spaces and in certain Banach spaces; see, for example, [18,8,17, 20,5, 30,3,6] and the references therein.

Recall that a mapping T :K!H is calledL-Lipschitzian if for allx; y2K;

kT x T yk Lkx yk;

whereL 0is a constant. In particular, ifL2[0;1) thenT is called a contraction mapping; ifL= 1 then T is called a nonexpansive mapping. A point x2K is called a …xed point of T ifT x =x:We denote the set of all …xed points ofT byF ix(T): A mappingT is said to be

(1) quasi-nonexpansive if F ix(T)6=; and

kT x pk kx pk; x2K; p2F ix(T);

(2) …rmly nonexpansive if for allx; y2K; we have

kT x T yk² hT x T y; x yi:

For nonexpansive mappings with …xed points, Mann iterative method [23] is a valuable tool to study them. Mann’s scheme is de…ned by:

x02K;

xn+1= nxn+ (1 n)T xn;

wheref ngis a sequence in(0;1):But Mann’s iteration process has only weak convergence, even in Hilbert space setting. Hence the modi…cation is necessary in order to guarantee the strong convergence of Mann’s method. Lot of works have been done for the modi…cation of the Mann’s iteration so that strong convergence is guaranteed. See, e.g., [31,35,36,16,25,22,11, 12, 29,13,26] and the reference therein.

If T₁ and T₂ are self-mappings onK; a pointx 2 K is called a common …xed point ofT_i(i = 1;2) if x2F ix(T1)\F ix(T2):To …nd a solution of the common …xed point problems, several iterative approximation methods were introduced and studied. This problem can be applied in solving solutions of various problems

in science and applied science, see [15,21,9] for instance. For almost all the results on common …xed point of nonlinear mappings in Hilbert spaces, commuting assumptions are needed on the operators.

Now, we introduce two minimization problems coupled with …xed point problems, …rstly, we consider the constrained convex minimization problem coupled with …xed point problem involving two mappings, namely,

…nd anx with the property:

x 2 \F ix(T1)\F ix(T2): (3)

On the other hand, we consider the constrained convex minimization problem coupled with …xed point problem involving composed mapping, namely, …nd anx with the property:

x 2 \F ix(T1 T2); (4)

whereT1 andT2 be quasi-nonexpansive and …rmly nonexpansive mappings onK;respectively.

Remark 1 Easily, we obtain the following conclusions:

(i) F ix(T1)\F ix(T2) F ix(T1 T2);

(ii) Problem of …nding an element of \F ix(T₁ T₂)is more general and more complex than the problem of …nding an element of \F ix(T₁)\F ix(T₂):

Above discussion suggests the following questions.

Question 1: Is it always true that the set of solutions of problem (3) coincides with the set of solutions of problem (4) without commuting assumptions?

Question 2: Could we construct an explicit algorithm based on a modi…ed Mann iterative method and the gradient-projection algorithm such that it converges strongly to a solution of problem (4) without compactness assumption?

The purpose of this paper is to give a¢ rmative answers to these questions mentioned above. Applications are also considered.

2 Preliminaries

Recall that a mapA:H !H;the domain ofA; D(A);the image of a subsetS ofH; A(S)the range ofA, R(A)and the graph ofA; G(A)are de…ned as follows:

D(A) :=fx2 H : Ax6=;g; A(S) :=[fAx : x2 Sg; R(A) :=A(H); G(A) :=f[x; u] : x2 D(A); u2 Axg:

LetK be a nonempty, closed and convex subset ofH:An operatorA:K!H is calledmonotone if hAx Ay; x yi 0; 8 x; y2K:

An operatorA:K!H is said -inverse strongly monotone if there exists a constant >0 such that hAx Ay; x yi kAx Ayk²; 8x; y2K:

The demiclosedness of a nonlinear operatorTusually plays an important role in dealing with the convergence of …xed point iterative algorithms.

De…nition 1 Let H be a real Hilbert space and T : D(T) H ! H be a mapping. I T is said to be demiclosed at 0 if for any sequence fx_ng D(T) such that fx_ng converges weakly to p and kx_n T x_nk converges to zero, thenp2F ix(T):

Lemma 2 ([4]) Let H be a real Hilbert space, K be a closed convex subset of H, and T : K ! K be a nonexpansive mapping such that F(T)6=;. Then I T is demiclosed.

Lemma 3 ([10]) Let H be a real Hilbert space. Then for anyx; y2H; the following inequalities hold:

kx+yk² kxk²+ 2hy; x+yi;

k x+ (1 )yk²= kxk²+ (1 )kyk² (1 ) kx yk²; 2(0;1):

Lemma 4 ( [34]) Assume that fang is a sequence of nonnegative real numbers such that an+1 (1

n)an+ n n for alln 0;wheref ⁿg is a sequence in(0;1) andf ⁿg is a sequence inRsuch that (a) P₁

n=0 n=1,

(b) lim sup_{n!1 n} 0 orP₁

n=0j n nj<1. Thenlim_n!1a_n= 0.

Lemma 5 ([24]) Lettn be a sequence of real numbers that does not decrease at in…nity in a sense that there exists a subsequencetni oftn such thattni such thattni tni+1 for alli 0: For su¢ ciently large numbers n2N; an integer sequencef (n)g is de…ned as follows:

(n) = maxfk n: tk tk+1g: Then, (n)! 1 asn! 1and

maxft _(n); tng t _(n)+1:

Lemma 6 Let H be a real Hilbert space and K be a nonempty, closed convex subset ofH: Let A:K!H be an -inverse strongly monotone mapping. Then, I A is nonexpansive mapping for all x; y2 K and

2[0;2 ]:

Proof. For allx; y2K;we have

k(I A)x (I A)yk² = k(x y) (Ax Ay)k²

= kx yk² 2 hAx Ay; x yi+ ²kAx Ayk² By using property ofAand 2[0;2 ];we have

k(I A)x (I A)yk²=kx yk²+ ( 2 )kAx Ayk² kx yk²: This shows thatI Ais nonexpansive.

Lemma 7 ([2]) Let H be a real Hilbert space,g a continuously Fréchet di¤ erentiable, convex functional on H andrg the gradient of g:If rg is ¹-Lipschitz continuous, then rg is -inverse strongly monotone.

Lemma 8 Let H be a real Hilbert space and K be a nonempty, closed convex subset of H: Let g :K!R be a continuously Fréchet di¤ erentiable, convex functional on K with a 1

-Lipschitz continuous rg: Then, I rg is nonexpansive mapping for allx; y2K and 2[0;2 ]:

Proof. The proof follows Lemmas6 and7.

Remark 2 A necessary condition of optimality for a point x 2 is that x 2V I(rg; K);where

V I(rg; K) :=fx 2K; hrg(x ); x x i 0; 8x2Kg:

Lemma 9 ([18]) Let K be a nonempty closed convex of a real HilbertH:Let g:K!R be a continuously Fréchet di¤ erentiable, convex functional onK with a 1

-Lipschitz continuous rg: Then for all >0;

V I(rg; K) =F ix(P_K(I rg)):

3 Main Results

We start by the following result.

Lemma 10 Let H be a real Hilbert space and let K be a nonempty closed convex subset of H: Let T1 : K ! K be a quasi-nonexpansive mapping and T2 : K ! K be a …rmly nonexpansive mapping. Then, F ix(T1)\F ix(T2) =F ix(T1 T2)andT1 T2 is a quasi-nonexpansive mapping onK:

Proof. We split the proof into two steps.

Step 1: First, we show thatF ix(T1)\F ix(T2) =F ix(T1 T2):We note that F ix(T1)\F ix(T2) F ix(T1 T2):

Thus, we only need to show that F ix(T1 T2) F ix(T1)\F ix(T2): Let p 2 F ix(T1)\F ix(T2) and q2F ix(T1 T2): By using properties ofT1 andT2, we have

kq pk²=kT1 T2q T1pk² kT2q pk²: (5) Using the fact thatT2 is …rmly nonexpansive, we have

kT₂q pk² hT₂q p; q pi

= 1

2(kT2q pk²+kq pk² kT2q qk²: (6) By virtue of (6), we can infer that

kT2q pk² kq pk² kT2q qk²: (7) Using (5) implies that (7) becomes

kT2q pk² kq pk² kT2q qk² kT2q pk² kT2q qk²: Clearly,kT2q qk= 0which implies that

q=T2q:

Keeping in mind thatT1 T2q=q; we have

q=T1 T2q=T1q:

Thus,q2F ix(T1)\F ix(T2):Hence,F ix(T1)\F ix(T2) =F ix(T1 T2):

Step 2: We show T1 T2 is a quasi-nonexpansive mapping on K:Letx2K andp2F ix(T1 T2): Then, p2F ix(T1)\F ix(T2)by step 1. We observe that,

kT1 T2x pk=kT1 T2x T1pk kT2x pk kx pk: This completes the proof.

We now prove the following theorem.

Theorem 11 Let H be a real Hilbert space and K a nonempty, closed convex cone of H: Let g : K !R be a continuously Fréchet di¤ erentiable, convex functional on K with a 1

-Lipschitz continuous rg: Let T1:K!K be a quasi-nonexpansive mapping and T2:K!K be a …rmly nonexpansive mapping such that := \F ix(T1)\F ix(T2)6=;: Assume thatI T1 T2 is demiclosed at origin and 2(0;2 ): Letfxng be a sequence generated iteratively from arbitraryx02K by

8>

<

>:

zn= nxn+ (1 n)T1 T2xn; y_n = _nz_n+ (1 _n)P_K(I rg)z_n; xn+1= n( nxn) + (1 n)yn;

wheref ⁿg;f ⁿg; f ⁿg andf ng be sequences in(0;1):Suppose the following conditions hold:

(i) lim_{n!1 n}= 0;lim_{n!1 n} = 1andP₁

n=0(1 _n) _n =1; (ii) lim infn!1 n(1 n)>0;

(iii) lim infn!1 n(1 _n)>0:

Then, the sequencefxng converges strongly to x 2 ;wherex =P (0):

Proof. First of all, we prove that the sequencefxngis bounded. Indeed, if we letp2 ;by using Lemma3, we get

kzn pk² = n(xn p) + (1 n)(T1 T2xn p)

2

= _nkx_n pk²+ (1 _n)kT₁ T₂x_n pk² n(1 _n)kT₁ T₂x_n x_nk² kxn pk^{2 n}(1 n)kT1 T2xn xnk²:

Since _n 2(0;1);we get that

kz_n pk² kx_n pk²: (8)

By using the de…nition offxngand Lemma 8, it follows that

ky_n pk = k nz_n+ (1 _n)P_K(I rg)z_n pk

nkzn pk+ (1 _n)kPK(I rg)zn pk

kz_n pk: (9)

From (8) and (9), we have

kyn pk kzn pk kxn pk: (10)

From (10), we have

kxn+1 pk = k ⁿ( nxn) + (1 n)yn pk

n nkx_n pk+ (1 _n)ky_n pk+ (1 _n) _nkpk

n nkxn pk+ (1 n)kxn pk+ (1 n) nkpk [1 (1 n) n]kxn pk+ (1 n) nkpk

maxfkx_n pk; kpkg: By induction, we get

kxn pk maxfkx0 pk; kpkg; n 1:

Then, we obtain thatfxng is bounded, and so arefyng;fzng: By (8) and convexity of k:k², we obtain kxn+1 pk² k ⁿ( nxn) + (1 n)yn pk²

nk( _nx_n) pk²+ (1 _n)ky_n pk²

nk( nxn) pk²+ (1 n)kzn pk²

nk( nxn) pk²+ (1 n) h

kxn pk²

n(1 n)kT1 T2xn xnk² i

: Thus,

(1 n)(1 n) nkT1 T2xn xnk² kxn pk² kxn+1 pk²+ nk( nxn) pk²: Hence,

(1 n)(1 n) nkT1 T2xn xnk² kxn pk² kxn+1 pk²+ nk( nxn) pk²: (11)

Now we divide the rest of the proof into two cases.

Case I. Assume that there isn02N such thatfkxn pkgis decreasing for alln n0:Sincefkxn pkg is monotonic and bounded,fkxn pkgis convergent. Clearly, we have

nlim!1

h

kx_n pk² kx_n+1 pk²i

= 0:

It then implies from (11) that

nlim!1(1 n) nkT1 T2xn xnk²= 0: (12) Since _n 2(0;1)andlim inf_{n!1 n}(1 _n)>0;we have

nlim!1 x_n T₁ T₂x_n = 0: (13)

Now, we observe that,

kzn xnk = k(1 n)xn+ nT1 T2xn xnk

= k(1 _n)x_n+ _nT₁ T₂x_{n n}x_n (1 _n)x_nk kT1 T2xn xnk:

Therefore, from (13) we have

nlim!1kzn xnk= 0: (14)

Then from Lemma11, inequality (8) and the fact thatPK(I rg)is nonexpasive, we have ky_n pk² = k nz_n+ (1 _n)P_K(I rg)z_n pk²

= _nkzn pk²+ (1 _n)kPK(I rg)zn pk²

n(1 _n)kP_K(I rg)z_n z_nk²

kxn pk² n(1 _n)kPK(I rg)zn znk²: (15) By de…ntion offxng and the above inequality, we get

kxn+1 pk² = k ⁿ( nxn) + (1 n)yn pk²

nk( _nx_n) pk²+ (1 _n)ky_n pk²

nk( nxn) pk²+ (1 n)(kxn pk²

n(1 _n)kP_K(I rg)z_n z_nk²):

Thus,

(1 n) _n(1 _n)kPK(I rg)zn znk² kxn pk² kxn+1 pk² + nk( nxn) pk²: Since _n 2(0;1)andlim inf_{n!1 n}(1 _n)>0;we have

nlim!1kP_K(I rg)z_n z_nk= 0: (16)

SinceH is re‡exive andfxngis bounded, there exists a subsequencefxnkgoffxng such thatxnk converges weakly toainK and

lim sup

n!+1hx ; x xni= lim

k!+1hx ; x xnki:

From (13) and I T₁ T₂ is demiclosed, we obtain a 2 F ix(T₁ T₂): Using Lemma 10, we have a 2 F ix(T2)\F ix(T1):It follows from (16) and Lemma 2, we obtaina2F ix(PK(I rg)):By Lemma9, we havea2V I(rg; K):Therefore,a2 :On other hand, using property ofx (x =P (0));we then have

lim sup

n!+1hx ; x x_ni = lim

k!+1hx ; x x_nki

= hx ; x ai 0:

Finally, we show thatxn!x :Applying Lemma3, we get

kxn+1 x k² = hxn+1 x ; xn+1 x i= n nhxn x ; xn+1 x i

+(1 _n) _nhx ; x x_n+1i+ (1 _n)hy_n x ; x_n+1 x i

n nhxn x ; xn+1 x i+ (1 n) nhx ; x xn+1i +(1 _n)ky_n x kkx_n+1 x k

n nkxn x kkxn+1 x k+ (1 n) nhx ; x xn+1i +(1 n)kxn x kkxn+1 x k

[1 (1 _n) _n]kx_n x kkx_n+1 x k +(1 n) nhx ; x xn+1i

1 (1 n) n

2 (kxn x k²+kxn+1 x k²) +(1 _n) _nhx ; x x_n+1i;

which implies that

kx_n+1 x k² [1 (1 _n) _n]kx_n x k+ 2(1 _n) _nhx ; x x_n+1i: We can check that all the assumptions of Lemma4 are satis…ed. Therefore, we deducexn !x :

Case II. Assume that the sequencefkxn x kgis not monotonically decreasing. SetBn=kxn x kand :N!Nbe a mapping for alln (for somen0large enough) by (n) = maxfk2N:k n; Bk Bk+1g: We have is a non-decreasing such that (n)! 1 as n! 1andB _(n) B _(n)+1 forn n0:From (11), we have

(1 _(n))(1 _(n)) _(n)kx _(n) T1 T2x _(n)k² (n)k (n)x _(n) x k²!0 as n! 1: Since _n 2]0;1[andlim inf_n!1(1 _(n)) _(n)>0;we can deduce

nlim!1kx _(n) T1 T2x _(n)k= 0: (17) By a similar argument as in case 1, we can show thatx _(n)converges weakly inHandlim sup

n!+1hx ; x x _(n)i 0: We have for alln n0;

0 kx _(n)+1 x k² kx _(n) x k² 1 _{(n) (n)}[ kx _(n) x k²+ 2hx ; x x _(n)+1i];

which implies that

kx _(n) x k² 2hx ; x x _(n)+1i: Then, we have

nlim!1kx _(n) x k²= 0:

Therefore,

nlim!1B _(n)= lim

n!1B _(n)+1= 0:

Thus, by Lemma5, we conclude that

0 Bn maxfB _(n); B _(n)+1g=B _(n)+1: Hence, lim

n!1B_n= 0;that is fx_ngconverges strongly tox :This completes the proof.

We now apply Theorem 11 when T1 is a nonexpansive mapping. In this case demiclosedness assump- tion (I T1 T2 is demiclosed at origin) is not necessary.

Theorem 12 Let H be a real Hilbert space and K a nonempty, closed convex cone of H: Let g : K !R be a continuously Fréchet di¤ erentiable, convex functional on K with a 1

-Lipschitz continuous rg: Let T1 : K ! K be a nonexpansive mapping and T2 : K ! K be a …rmly nonexpansive mapping such that := \F ix(T1)\F ix(T2)6=;: Assume that 2(0;2 ): Letfxngbe a sequence generated iteratively from arbitraryx₀2K by 8

><

>:

z_n= _nx_n+ (1 _n)T₁ T₂x_n; yn = _nzn+ (1 _n)PK(I rg)zn; x_n+1= _n( _nx_n) + (1 _n)y_n;

wheref ng;f ng; f ng andf ng be sequences in(0;1):Suppose the following conditions hold:

(i) lim_{n!1 n}= 0;lim_{n!1 n} = 1andP₁

n=0(1 _n) _n =1; (ii) lim infn!1 n(1 n)>0;

(iii) lim inf_{n!1 n}(1 _n)>0:

Then, the sequencefxng converges strongly to x 2 wherex =P (0):

Proof. We haveT1 T2is nonexpansive mapping, then, the proof follows Lemma2and Theorem 11.

IfTi I;fori= 1;2; then Theorem11is reduced to the following:

Theorem 13 Let H be a real Hilbert space and K a nonempty, closed convex cone of H: Let g : K !R be a continuously Fréchet di¤ erentiable, convex functional onK with a 1

-Lipschitz continuousrg:Suppose that the minimization problem (1)is consistent and 2(0;2 ):Letfx_ngbe a sequence generated iteratively from arbitrary x₀2K by: (

yn= _nxn+ (1 _n)PK(I rg)xn; xn+1= n( nxn) + (1 n)yn;

wheref ng;f ng andf ng be sequences in(0;1):Suppose the following conditions hold:

(i) lim_{n!1 n}= 0;lim_{n!1 n} = 1andP₁

n=0(1 _n) _n =1; (ii) lim infn!1 n(1 _n)>0:

Then, the sequencefx_ng converges strongly to a minimizer ofg:

Ifg 0; then Theorem11is reduced to the following:

Theorem 14 LetH be a real Hilbert space andK a nonempty, closed convex cone ofH:Let T1:K!Kbe a nonexpansive mapping andT2:K!Kbe a …rmly nonexpansive mapping such thatF ix(T1)\F ix(T2)6=;: Let fxng be a sequence de…ned as follows:

8>

<

>:

z_n = _nx_n+ (1 _n)T₁ T₂x_n, yn= _nzn+ (1 _n)Pk(I rg)zn; x_n+1= _n( _nx_n) + (1 _n)y_n;

wheref ⁿg;f ⁿg andf ⁿg be sequences in(0;1):Suppose the following conditions hold:

(i) lim_{n!1 n}= 0;lim_{n!1 n} = 1andP₁

n=0(1 _n) _n =1; (ii) lim infn!1 n(1 n)>0:

Then, the sequencefxng converges strongly to a common …xed point of T1 andT2:

Remark 3 In our theorems, we assume that K is a cone. But, in some cases, for example, if K is the closed unit ball, we can weaken this assumption to the following: x 2 K for all 2 (0;1) and x 2 K:

Therefore, our results can be used to approximate a common solution of convex minimization problem and

…xed point problem involving composed operators from the closed unit ball to itself.

4 Application to Some Nonlinear Problems

In this section, we apply our main results for …nding a common solution of …xed points problem, convex minimization problem and equlibrium problem.

Problem 1 LetK be a nonempty, closed convex subset of a real Hilbert spaceH:We consider the following minimization problem :

xmin2Kg(x); (18)

whereg be a continuously Fréchet di¤ erentiable, convex functional onK:

We denote the set of solutions of Problem 1by 1:

Problem 2 LetK be a nonempty, closed convex subset of a real Hilbert spaceH:We consider the following

…xed point problem :

…ndx2Ksuch thatx=T x; (19)

whereT :K!K be a quasi-nonexpansive mapping.

We denote the set of solutions of Problem 2by ₂:

Problem 3 LetG:K K!Rbe a bifunction whereRis the set of real numbers. The equilibrium problem corresponding toGis to …nd x 2K such that

G(x ; y) 0;8y2K: (20)

The set of solutions of Problem 3 is denoted by EP(G): Numerous problems in physics, optimization, and economics are reduced to …nd the solution of an equilibrium problem (e.g., see [18]). For solving the equilibrium problem we assume that the bifunctionGsatis…es the following conditions:

(A1) G(x; x) = 0for allx2K;

(A2) Gis monotone, i.e.,G(x; y) +G(y; x) 0 for allx; y2K;

(A3) for eachx; y; z2K;

tlim!0G(tz+ (1 t)x; y) G(x; y);

(A4) for eachx2K; y!G(x; y)is convex and lower semicontinuous.

For solving Problem 3, we introduce the following lemma.

Lemma 15 ([32]) Assume thatG:K K!Rsatis…es(A1)-(A4):Forr >0andx2H;de…ne a mapping T_r^G:H !K as follows

T_r^G(x) =fz2K; G(z; y) +1

rhy z; z xi 0; 8y2Kg; for allx2H:Then, the following hold:

1. T_r^G is single-valued;

2. T_r^G is …rmly nonexpansive;

3. F ix(T_r^G) =EP(G);

4. EP(G) is closed and convex.

Therefore, by Theorem11, the following result is obtained.

Theorem 16 LetH be a real Hilbert space and K a nonempty, closed convex cone of H:Let g:K!Rbe a continuously Fréchet di¤ erentiable, convex functional on K with a 1

-Lipschitz continuous rg such that 2(0;2 )and, let T :K!K be a quasi-nonexpansive mapping. LetG be a bifunction fromK K!R satis…es(A1)-(A4) Such that 1\ ²\EP(G)6=; andI T T^G is demiclosed at origin. Let fxng be a sequence de…ned as follows: 8

>>

><

>>

>:

x₀2K; choosen arbitrarily, zn= nxn+ (1 n)T T^Gxn, y_n = _nz_n+ (1 _n)P_K(I rg)z_n; xn+1= n( nxn) + (1 n)yn;

wheref ⁿg;f ⁿg; f ⁿg andf ng be sequences in(0;1):Suppose the following conditions hold:

(i) limn!1 n= 0;limn!1 n = 1andP₁

n=0(1 n) n=1; (ii) lim infn!1 n(1 n)>0;

(iii) lim infn!1 n(1 _n)>0:

Then, the sequencesfxngconverges a strongly to common solution of Problem 1, Problem2and Problem3.

5 Open Problems

In this paper, we have only shown that F ix(T₁ T₂) = F ix(T₁)\F ix(T₂) with T₁ and T₂ are quasi- nonexpansive and …rmly nonexpansive mappings respectively. It is well known that there are other nonlinear mappings more general than …rmly nonexpansive mappings and quasi-nonexpansive mappings. Therefore, the results of this paper open up many forthcoming results regarding convex minimization problem coupled with the …xed point problem studied in this paper. These following questions are open for researchers interested in this …eld:

(i) Can we extend Lemma 10 to mappings that are more general than …rmly nonexpansive and quasi- nonexpansive mappings mappings ?

(ii) Do the results hold in the setting of a more general Banach space by using our algorithm de…ned in Theorem11?

References

[1] K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces,Fixed Point Theory Appl. 2006, Art. ID 35390, 13 pp.

[2] J. B. Baillon and G. Haddad, Quelques proprits des oprateurs angle-borns et n-cycliquement monotones, Israel J. Math., 26(1977), 137–150.

[3] H. H. Bauschke, J. V. Burke, F. R. Deutsch, H. S. Hundal and J. D. Vanderwer¤, A new proximal point iteration that converges weakly but not in norm,Proc. Amer. Math. Soc., 133(2005), 1829–1835.

[4] F. E. Browder, Convergenge theorem for sequence of nonlinear operator in Banach spaces, Math. Z., 100(1967), 201–225.

[5] G. Cai and Y. Shehu, An iterative algorithm for …xed point problem and convex minimization problem with applications, Fixed Point Theory Appl., 2015, 2015:7, 17 pp.

[6] L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient iterative methods for variational in- equalities, Appl. Math. Computation, 218(2011), 1112–1123.

[7] A. Cegieslki, Iterative Methods for Fixed Point Problems in Hilbert spaces, Lecture Note in Mathemat- ics, Springer-Verglag, Heidelberg, 2012.

[8] P. L. Combettes, A block-iterative surrogate constraint splitting method for quadratic signal recovery, IEEE Transactions on Signal Processing, 51(2003), 1771–1782.

[9] Y. J. Cho, H. Zhou and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47(2004), 707–717.

[10] C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer Verlag Series:

Lecture Notes in Mathematics, Vol. 1965,(2009), ISBN 978-1-84882-189.

[11] R. Dubey, Deepmala and V. N. Mishra, Higher-order symmetric duality in nondi¤erentiable multiob- jective fractional programming problem over cone constraints, Stat., Optim. Inf. Comput., 8(2020), 187–205.

[12] R. Dubey, Vandana and V. N. Mishra, Second-order multiobjective symmetric programming problem and duality relations under (F; Gf)-convexity, Global Journal of Engineering Science and Researches, 5(2018), 187–199.

[13] G. Farid, V. N. Mishra and S. Mehmood, Hadamard and Fejér-Hadamard type inequalities for convex and relative convex functions via an extended generalized mittag-le- er function, International Journal of Analysis and Applications, 17(2019), 892–903.

[14] A. Genel and J. Lindenstrass, An example concerning …xed points, Israel J. Math., 22(1975), 81–86.

[15] B. Gunduz and S. Akbulutl, Common …xed points of a …nite family of iasymptotically nonexpansive mappings by s-iteration process in banach spaces, Thai J. Math., 15(2017), 673–687.

[16] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 3(1967), 957–961.

[17] I. Hideaki, Convergence Analysis of Iterative Methods for Nonsmooth Convex Optimization over Fixed Point Sets, Mathematical Programming, October (2015).

[18] I. Hideaki, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program, 133(2012), 227–242.

[19] I. Hideaki, Fixed point optimization algorithm and its application to network bandwidth allocation, J.

Comput. Appl. Math., 236(2012), 1733–1742.

[20] I. Hideaki and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the …xed point set of a nonexpansive mapping, SIAMJ.Optim., 19(2009) 1881–1893.

[21] S. Kosol, Weak and strong convergence theorems of some iterative methods for common …xed point of Berinde nonexpansive mappings, Thai J. Math., 15(2017), 629–639.

[22] X. Liu, M. Zhou, L. N. Mishra, V. N. Mishra and B. Damjanovi´c, Common …xed point theorem of six self-mappings in Menger spaces using(CLRST)property, Open Mathematics, 16(2018), 1423–1434.

[23] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4(1953), 506–510.

[24] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis, 16(2008), 899–912.

[25] L. N. Mishra, On existence and behavior of solutions to some nonlinear integral equations with appli- cations, Ph.D. Thesis (2017), National Institute of Technology, Silchar 788 010, Assam, India.

[26] V. N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India.

[27] B. T. Polyak, Introduction to Optimization Software, New York, 1987.

[28] A. Ruszczynski, Nonlinear Optimization, Princeton University Press, New Jersey, 2006.

[29] A. Ur. Rehman, G. Farid and V. N. Mishra, Generalized convex function and associated Petrovi´c’s inequality, Int. J. Anal. Appl., 17(2019), 122–131.

[30] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common …xed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ricerche Mat., 69(2019), 235–259.

[31] T. M. M. Sow, N. Djitté and C.E. Chidume, A path convergence theorem and construction of …xed points for nonexpansive mappings in certain Banach spaces, Carpathian J. Math., 32(2016), 241–250.

[32] H. K. Xu, A variable Krasnoselskii-Mann algorithm and the multiple set split feasiblity problem, Inverse Problem, 26(2006), 2021–2034.

[33] H. K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., 150(2011), 360–378.

[34] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66(2002), 240–256.

[35] Y. Yao and Y. Je. Cho, Strong convergence of modi…ed Krasnoselskii-Mann iterative algorithm for nonexpansive mappings in Hilbert Spaces, Math. Model. Anal., 15(2010), 265–274.

[36] Y. Yao, H. Zhou and Y. C. Liou, Strong convergence of modi…ed Krasnoselskii-Mann iterative algorithm for nonexpansive mappings, J. Math. Anal. Appl. Comput., 29(2009), 383–389.